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A321585
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Number of connected nonnegative integer matrices with sum of entries equal to n and no zero rows or columns.
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6
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1, 1, 3, 11, 52, 312, 2290, 19920, 200522, 2293677, 29389005, 416998371, 6490825772, 109972169413, 2014696874717, 39684502845893, 836348775861331, 18777970539419957, 447471215460930665, 11279275874429302811, 299844572529989373703, 8383794111721619471384, 245956060268568277412668
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OFFSET
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0,3
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COMMENTS
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A matrix is connected if the positions in each row (or each column) of the nonzero entries form a connected hypergraph.
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LINKS
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EXAMPLE
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The a(3) = 11 matrices:
[3] [2 1] [1 2] [1 1 1]
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[2] [1 1] [1 1] [1] [1 0] [0 1]
[1] [1 0] [0 1] [2] [1 1] [1 1]
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[1]
[1]
[1]
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MATHEMATICA
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multsubs[set_, k_]:=If[k==0, {{}}, Join@@Table[Prepend[#, set[[i]]]&/@multsubs[Drop[set, i-1], k-1], {i, Length[set]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[multsubs[Tuples[Range[n], 2], n], And[Union[First/@#]==Range[Max@@First/@#], Union[Last/@#]==Range[Max@@Last/@#], Length[csm[Map[Last, GatherBy[#, First], {2}]]]==1]&]], {n, 5}] (* Mathematica 7.0+ *)
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PROG
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(PARI)
NonZeroCols(M)={my(C=Vec(M)); Mat(vector(#C, n, sum(k=1, n, (-1)^(n-k)*binomial(n, k)*C[k])))}
ConnectedMats(M)={my([m, n]=matsize(M), R=matrix(m, n)); for(m=1, m, for(n=1, n, R[m, n] = M[m, n] - sum(i=1, m-1, sum(j=1, n-1, binomial(m-1, i-1)*binomial(n, j)*R[i, j]*M[m-i, n-j])))); R}
seq(n)={my(M=matrix(n, n, i, j, sum(k=1, n, binomial(i*j+k-1, k)*x^k, O(x*x^n) ))); Vec(1 + vecsum(vecsum(Vec( ConnectedMats( NonZeroCols( NonZeroCols(M)~))))))} \\ Andrew Howroyd, Jan 17 2024
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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